Find the smallest, positive four-digit multiple of $15.$
Explanation: Dividing $1000$ by $15$ gives a quotient of $66$ with a remainder of $10.$ In other words, \[1000=15\cdot66+10.\]So, $66\cdot15$ is the largest three-digit multiple of $15,$ and $67\cdot15=\boxed{1005}$ is the smallest four-digit multiple.